What is the medium of the content?
If they are videos, then YouTube would probably be first option but if they are, say, pdfs, then you might have to make a blog and then upload to that there. It still might feel like an island at first but it will gain traffic from your students and then other people who find it in search results. There are also problem-...
Here are some ideas:
As Brendan says above, you can give an example or non-example, and ask the student to explain why (using the definition) it is an example or non-example.
You can ask for examples satisfying certain properties ("Give an example of a relation on a three element set which is transitive and symmetric, but not reflexive").
You can ...
I'll reiterate my recommendation of MyOpenMath. To your points:
It understands TeX (a simplified version, using ` math ` instead of \$ math \$)
It sllows the use of parameters in some way to randomize numbers for each student
It allows these parameters within the TeX code.
It can be integrated into Blackboard (that's how I've been using it) using an LTI ...